# Thermodynamics - Calorific Capacity

1. Nov 25, 2012

### Jalo

1. The problem statement, all variables and given/known data

Consider a system with a constant number of particles.
Write the total differential dS in terms of the derivarives $\frac{∂S}{∂T}$ and $\frac{∂S}{∂V}$. Introduce CV (calorific capacity at constant volume).
Next write the total differential of the volume dV in terms of the parcial derivatives $\frac{∂V}{∂T}$ and $\frac{∂V}{∂P}$. Assume that the pressure is constant. Show that the result comes in the form of:

CP-CV= Expression

2. Relevant equations

CP=T$\frac{∂S}{∂T}$ , P and N Constant
CV=T$\frac{∂S}{∂T}$ , V and N Constant

3. The attempt at a solution

First I wrote the differential of the entropy as asked:

dS = $\frac{∂S}{∂T}$dT + $\frac{∂S}{∂V}$dV

I know that $\frac{∂S}{∂V}$ = CV/T. Substituting I get:

dS = CV/T dT + $\frac{∂S}{∂V}$dV

Next I found the differential of the volume:

dV = $\frac{∂V}{∂T}$dT + $\frac{∂V}{∂P}$dP

Since the pressure is constant it reduces to the form

dV = $\frac{∂V}{∂T}$dT

Substituting in our dS expression we get:

dS = CV/T dT + $\frac{∂S}{∂V}$$\frac{∂V}{∂T}$dT =
= CV/T dT + $\frac{∂S}{∂T}$dT =
= CV/T dT + CP/T dT ⇔
⇔ T$\frac{dS}{dT}$ = CV + CP

I'm making some mistake. If anyone could point me in the right direction I'd appreciate.

Thanks!

2. Nov 25, 2012

### haruspex

I think your problem is a confusion over the meanings of different partial derivatives. Your use of $\frac{∂S}{∂V}$ refers to changing volume, keeping temperature constant.
That's inconsistent with later equating $\frac{∂S}{∂V}\frac{∂V}{∂T} = \frac{∂S}{∂T}$, which, to be valid, assumes pressure constant in all terms.